At times, finding sufficient evidence based on personal observation to prove a claim for relief is a difficult task, as difficult as that facing a farsighted tailor attempting to thread a needle: what’s at hand will just not do the job. But unlike the dilemma in which the tailor finds his wandering fingers, the dilemma in which litigators find themselves has a practical solution: the doctrine of “circumstantial evidence.” Circumstantial evidence is, of course, an inference from what has been observed accepted as proof of what is unobserved. For instance, the circumstance of Macbeth with a bloody dagger in his quaking hand, standing over the punctured, lifeless body of Duncan generates the inference that Macbeth has just slain Duncan.

This example is an instance of what might be called “run of the mill” circumstantial evidence. Sometimes, however, also available are “specially milled” doctrines of circumstantial evidence, such as the doctrine of res ipsa loquitur; the use of differential diagnosis to establish general causation (designed to sanction an inference about a law of nature from a case study); and the doctrine of indeterminate defect. These specially milled doctrines of circumstantial evidence accommodate inferences significantly “underdetermined” by the circumstances from which they were drawn, when sanctioning those weak inferences may “offer the plaintiff the only fair opportunity to recover.”

Of particular interest is the doctrine of “indeterminate defect.” This doctrine provides that the plaintiff’s harm was caused by a product defect existing at the time of sale or distribution, without proof of a specific defect, when the incident causing that harm (1) was of a kind that ordinarily occurs as a result of a product defect and (2) was not, in the particular case, solely the result of causes other than a product defect existing at the time of sale or distribution. Vanek v. Kirby, 253 Or 494, 450 P2d 778 (1969); Restatement (Third) of Torts: Products Liability § 3 (1998).

The criteria of this doctrine are a bit fuzzy, sparking endless debate about how far the doctrine should be stretched. Some urge that the doctrine be limited to new products destroyed in the accident. For instance, it applies to the new single-engine airplane that on its maiden voyage is seen to burst into flames, to crash, and then to sink to the bottom of the Newfoundland Basin, never to be seen again. Others would have it apply whenever a product fails to meet a consumer’s reasonable expectations. For instance, it applies to the automobile with 10,000 miles on the odometer whose tire shreds on a cross-country trip to Disney World.

As these examples signal, the degree to which the inferences are underdetermined by their foundations exists on a continuum. And along that continuum, no bright line distinguishes those foundations that are good enough to support an inference from those that are not good enough. As a result, under the traditional common law analysis, the process for legitimizing the inferences under this doctrine appears to be highly subjective.

Enter the clarifying theorem of that clever mathematical hobbyist, the Reverend Thomas Bayes (1702-1761). Bayes’ Theorem is a fundamental tool of inductive inference, providing a systematic way to reason backward from effects or harms to their likely causes. It is, symbolically:

p(H|e) = p(e|H) · p(H)
p(e)

P(H|e) is the probability of the hypothesis (for example, the car was defective) assuming that the evidence for that hypothesis is true (for instance. the car abruptly left the roadway). This variable is sometimes referred to as the “posterior probability.” P(e|H) is the probability of the evidence (for example, the car abruptly left the roadway) assuming that the hypothesis is true (for example, the car was defective). This variable is referred to as the “1ikelihood.” P(H) is the probability that the hypothesis is true before the evidence arises; it is the frequency or “base rate” of the hypothesis occurring before the evidence occurs (for example, before this particular accident, 20 percent of this manufacturer’s cars of this model had defects). And p(e) is merely what is called a normalizing constant equal to the probability of all events that could cause the evidence to occur. (Note: Probability variables take values from 0 to 1, with 0 representing no chance that an event will occur and 1 representing that the event is certain.) David Malakoff, Bayes Offers a “New” Way to Make Sense of Numbers, 286 Science 1460—64 (1999); CoIin Howson & Peter Urbach, Bayesian Reasoning in Science, 350 Science 371-74 (1991).

Bayes’ Theorem, applied to the doctrine of indeterminate defect, helps focus the inquiry under that doctrine. More particularly, the theorem provides an heuristic or framework within which evidence is appraised to determine whether one or another inferred hypothesis from that evidence is either reasonable or unreasonable. For instance, suppose that a relatively new car swerves unexpectedly from the roadway, resulting in an accident that injures the driver. Obviously, that evidence (e) may be the result of any number of causes: for example, the car was defective when sold (H1); the car was defective not when sold but as a result of negligent repairs performed after it was sold (H2); the driver fell asleep at the wheel (H3); the driver struck a large rock on the roadway (H4); … .

What the doctrine of indeterminate defect requires is that p(e|H1 or H2) have a relatively high value. That is, “the incident causing the harm was of a kind that ordinarily occurs as a result of a product defect.” Note that it is not sufficient that either p(H1) or p(H2) have a high value. A product may be defective in any number of respects (H1 or H2). But only some or perhaps none of these kinds of defect may result in an incident of the kind resulting in the plaintiff’s harm (e). The doctrine also requires that p(e|H) x p(H2) + p(e|H3) x p(H3) + … have a significantly low value. That is, the incident causing the harm must not he solely the result of causes (H2, H3, H4) other than a product defect existing at the time of sale or distribution (H1).

In this calculus, Bayes’ Theorem highlights the importance of having a minimum amount of evidence to establish the two basic variables of the theorem: the “likelihood” [p(e|H) and the ‘base rate” [p(H)]. High values for both likelihood [p( e|H)] and base rate [p(H)] result in a compelling inference. Low values for either the likelihood or the base rate result in a highly unreliable inference. For instance, if, on the one hand, p(e|H) is .8 and p(H) is .9, then p(H|e) is a compelling .72 or 72%. If, on the other hand, p(e|H) is .2 and p(H) is .3, then p(H|e) is a mere .06 or 6%; or if p(e|H) is .2 and p(H) is 1.0, then p(H|e) is a mere .2; or if p(e|H) is 1 .0 and p(H) is .3, then p(H|e) is .3.

Now consider some examples that tease out some of the distinctive features of the doctrine of indeterminate defect. First, consider the example of Subtle, the monocular alchemist. Subtle lost an eye when a ‘philosopher’s stone” (a composite bolus of rare chemicals) blew up in his face after he had applied the brand-new stone, with the help of mortar and pestle, to a variety of substances scavenged from his laboratory. After his injury, Subtle immediately had his attorney, Surly, sue the seller of the stone, Mammon’s Alchemical Supply Company, Ltd., under a theory of strict products liability. And because the philosopher’s stone was completely consumed in the conflagration, Surly sought to prove Subtle’s claim with the aid of the doctrine of indeterminate defect.

Subtle’s injury can he presumed to have been caused by three possible events: first, the stone itself could have been defective (H1); second, the substrate to which the stone was added could have been defective (H2) or, third, mixing both together could have been volatile (H3). Whether the stone was defective [p(e|H1) or p(e|H3)] is unknown because it was never found and examined. Even so, the probability that the stone would explode is limited because no previous stone sold by Mammon had exploded. That is, p(e|H1) is low; assume it to be on the order of .10. Yet the history of philosopher stones sold by Mammon indicated that none had ever met consumer expectations; that is, none had been successful in turning any substrate into gold. So p(H1) was high; assume it to be on the order of 1.0. Even so, p(e|H1) x p(H1) was only .10. Whether the substrate was volatile [p(e|H2)] is also unknown. It too was never found and examined. Unfortunately, no history about its volatility [p(H2)] was available. So Surly could not rule out H2. And so, accounting for all this information, p(H1|e) is indeterminate; that is, no reasonable inference about H1 can be drawn from e.

This example reveals several important nuances about the doctrine of indeterminate defect. First, the hypothesis of causation (H1) is framed in general terms because no specific defect has been identified as the cause of the incident resulting in the harm—hence, the use of the doctrine of “indeterminate” defect. Second, both the value for the likelihood [p(e|H1)] and for the base rate [p(H1)] must he high to establish that a product with an indeterminate defect caused the observed harm. So the unit of analysis is p(H) x p(e|H), not just p(H) or p(e|H). That is, p(H) x p(e|H) will have a low value if either p(H) or p(e|H ) has a low value. As this example demonstrates, it is not enough that the value for p(H1) is high where H1 is framed as an unspecified defect in the product. The value for p(e|H1) must also be high; that is, an unspecified defect in the product (H1) must have a high probability of resulting in the kind of incident that resulted in the observed harm. Third, if insufficient information is available about the likelihood and base rate of alternative causes that need to be ruled out, then the probability that a defect in the product caused the harm cannot be determined. Charles A. Holloway, Decision Making Under Uncertainty (1979) (explains methods for assessing p(H) and p(e|H) subjectively using “assessment lotteries”).

As this example also indicates, a major task in applying the doctrine of indeterminate defect is ruling out alternative potential causes of the harm. That process is no different from that in medicine known as “differential diagnosis.” But, unfortunately for Subtle, if, initially, the alleged cause (H1) has a high base rate, but other substantially probable causes (H2, H3,…) cannot be reliably ruled out, differential diagnosis is unhelpful in isolating the alleged cause, H1. Note, Navigating Uncertainty: Gatekeeping in the Absence of Hard Science, 113 Harv L Rev 1467-85 (2000).

Next, consider the example of Francis Macomber, the nearsighted big-game hunter. Macomber, while hunting in Africa with his new rifle, a .505 Gibbs, crossed paths with an angry rhinoceros. In response, Macomber was seen to have leveled his rifle, to have aimed it at the rhino, and to have pulled the trigger, but no recoil was then seen. Shortly thereafter, the rhino struck and killed him. In the melee, the sling of Macomber’s rifle caught on the rhino’s horn and was thus carried away when the rhino fled the scene. Neither rifle nor rhino was seen again.

Macomber’s death is presumed to have been caused by one of three possible events: first, the rifle could have been defective, for example, a flawed firing mechanism (H1); second, Macomber could have failed to switch the safety of his rifle to the off position (H2); third, the chambered round or bullet could have been a dud (H3). The probabilities of hypotheses (H2) and (H3) were subsequently limited. Eyewitnesses believe they saw Macomber switch the safety to the off position, thereby reducing the probability of p(e|H2) · p(H2). Other rounds presumed to be from the same box as the round Macomber chambered were found not to be defective, thereby reducing the probability of p(e|H3) · p(H3). Whether the rifle was defective is unknown; it was never found, but given the unlikelihood of hypotheses H2 and H3 the probability that it was defective is presumed to he high. That is, if p(e) or p(H1) · p(e|H1) + p(H2) x p(e|H2) + p(H3) · p(e|H3)… equals 1, then to the extent the value of p(H2) · p(e(H2) + p(H3) · p(e|H3) … is reduced (ruled out), the value of p(H1) · p(e|H1) is increased. Assume that the value for p(e|H1) is on the order of .80. Historically. this brand of rifle has had problems; indeed, for that reason it was extremely unpopular with big-game hunters. That is. the value for p(H1) was high, on the order of .70. So the value for p(e|H1 ) · p(H1) is high, .56 or 56 percent.

This last example reveals the limited circumstance in which the doctrine of indeterminate defect (“DID”) applies: when the value for p(H1) x p(e|H1) is high and the value for p(H2 ) · p(e|H2) + p(H3,) · p(e|H3) is low. DID does not apply in other possible circumstances. As the foregoing discussion has revealed, although the doctrine of indeterminate defect is important in the administration of’ justice, it has a very limited application.